m2r – Macaulay2 in R

Overview

m2r is a new R package that provides a persistent connection between R and Macaulay2 (M2).

The package grew out of a collaboration at the 2016 Mathematics Research Community on algebraic statistics, funded by the National Science Foundation through the American Mathematical Society.

If you have a feature request, please file an issue!

Getting started

m2r is loaded like any other R package:

library(m2r)
# Loading required package: mpoly
#   Please cite m2r! See citation("m2r") for details.
#   M2 found in /Applications/Macaulay2-1.10/bin

When loaded, m2r initializes a persistent connection to a back-end Macaulay2 session. The basic function in R that accesses this connection is m2(), which simply accepts a character string that is run by the Macaulay2 session.

m2("1 + 1")
# Starting M2... done.
# [1] "2"

You can see the persistence by setting variables and accessing them across different m2() calls:

m2("a = 1")
# [1] "1"
m2("a")
# [1] "1"

You can check the variables defined in the M2 session with m2_ls():

m2_ls()
# [1] "a"

You can also check if variables exist with m2_exists():

m2_exists("a")
# [1] TRUE
m2_exists(c("a","b"))
# [1]  TRUE FALSE

Apart from the basic connection to M2, m2r has basic data structures and methods to reference and manipulate the M2 objects within R. For more on this, see the m2r internals section below.

Rings, ideals, and Grobner bases

m2r currently has basic support for rings (think: polynomial rings):

(QQtxyz <- ring("t", "x", "y", "z", coefring = "QQ"))
# M2 Ring: QQ[t,x,y,z], grevlex order

and ideals of rings:

(I <- ideal("t^4 - x", "t^3 - y", "t^2 - z"))
# M2 Ideal of ring QQ[t,x,y,z] (grevlex) with generators : 
# < t^4  -  x,  t^3  -  y,  t^2  -  z >

You can compute Grobner bases as well. The basic function to do this is gb():

gb(I)
# z^2  -  x
# z t  -  y
# -1 z x  +  y^2
# -1 x  +  t y
# -1 z y  +  x t
# -1 z  +  t^2

Perhaps an easier way to do this is just to list off the polynomials as character strings:

gb("t^4 - x", "t^3 - y", "t^2 - z")
# z^2  -  x
# z t  -  y
# -1 z x  +  y^2
# -1 x  +  t y
# -1 z y  +  x t
# -1 z  +  t^2

The result is an mpolyList object, from the mpoly package. You can see the M2 code by adding code = TRUE:

gb("t^4 - x", "t^3 - y", "t^2 - z", code = TRUE)
# m2rintgb00000003 = gb(m2rintideal00000003); gens m2rintgb00000003

You can compute the basis respective of different monomial orders as well. The default ordering is the one in the respective ring, which defaults to grevlex; however, changing the order is as simple as changing the ring.

ring("x", "y", "t", "z", coefring = "QQ", order = "lex")
# M2 Ring: QQ[x,y,t,z], lex order
gb("t^4 - x", "t^3 - y", "t^2 - z")
# t^2  -  z
# -1 t z  +  y
# -1 z^2  +  x

On a technical level, ring(), ideal(), and gb() use nonstandard evaluation rules. A more stable way to use these functions is to use their standard evaluation versions ring_(), ideal_(), and gb_(). Each accepts first a data structure describing the relevant object of interest first as its own object. For example, at a basic level this simply changes the previous syntax to

use_ring(QQtxyz)
poly_chars <- c("t^4 - x", "t^3 - y", "t^2 - z")
gb_(poly_chars)
# z^2  -  x
# z t  -  y
# -1 z x  +  y^2
# -1 x  +  t y
# -1 z y  +  x t
# -1 z  +  t^2

gb_() is significantly easier to code with than gb() in the sense that its inputs and outputs are more predictable, so we strongly recommend that you use gb_(), especially inside of other functions and packages.

As far as other kinds of computations are concerned, we present a potpurri of examples below.

Ideal saturation:

ring("x", coefring = "QQ")
# M2 Ring: QQ[x], grevlex order
I <- ideal("(x-1) x (x+1)")
saturate(I, "x") # = (x-1) (x+1)
# M2 Ideal of ring QQ[x] (grevlex) with generator : 
# < x^2  -  1 >

Radicalization:

I <- ideal("x^2")
radical(I)
# M2 Ideal of ring QQ[x] (grevlex) with generator : 
# < x >

Primary decomposition:

ring("x", "y", "z", coefring = "QQ")
# M2 Ring: QQ[x,y,z], grevlex order
I <- ideal("x z", "y z")
primary_decomposition(I)
# M2 List of ideals of QQ[x,y,z] (grevlex) : 
# < z >
# < x,  y >

Dimension:

ring("x", "y", coefring = "QQ")
# M2 Ring: QQ[x,y], grevlex order
I <- ideal("y - (x+1)") 
dimension(I)
# [1] 1

Factoring integers and polynomials

You can compute prime decompositions of integers with factor_n():

(x <- 2^5 * 3^4 * 5^3 * 7^2 * 11^1)
# [1] 174636000
factor_n(x)
# $prime
# [1]  2  3  5  7 11
# 
# $power
# [1] 5 4 3 2 1

You can also factor polynomials over rings using factor_poly():

factor_poly("x^4 - y^4")
# $factor
# x  -  y
# x  +  y
# x^2  +  y^2
# 
# $power
# [1] 1 1 1

Smith normal form of a matrix

The Smith normal form of a matrix M here refers to the decomposition of an integer matrix D = PMQ, where D, P, and Q are integer matrices and D is diagonal. P and Q are unimodular matrices (their determinants are -1 or 1), so they are invertible. This is somewhat like a singular value decomposition for integer matrices.

M <- matrix(c(
   2,  4,   4,
  -6,  6,  12,
  10, -4, -16
), nrow = 3, byrow = TRUE)

(mats <- snf(M))
# $D
#      [,1] [,2] [,3]
# [1,]   12    0    0
# [2,]    0    6    0
# [3,]    0    0    2
# M2 Matrix over ZZ[] 
# 
# $P
#      [,1] [,2] [,3]
# [1,]    1    0    1
# [2,]    0    1    0
# [3,]    0    0    1
# M2 Matrix over ZZ[] 
# 
# $Q
#      [,1] [,2] [,3]
# [1,]    4   -2   -1
# [2,]   -2    3    1
# [3,]    3   -2   -1
# M2 Matrix over ZZ[]
P <- mats$P; D <- mats$D; Q <- mats$Q

P %*% M %*% Q                # = D
#      [,1] [,2] [,3]
# [1,]   12    0    0
# [2,]    0    6    0
# [3,]    0    0    2
solve(P) %*% D %*% solve(Q)  # = M
#      [,1] [,2] [,3]
# [1,]    2    4    4
# [2,]   -6    6   12
# [3,]   10   -4  -16

det(P)
# [1] 1
det(Q)
# [1] -1

m2r internals: pointers, reference and value functions, and m2 objects

At a basic level, m2r works by passing strings between R and M2. Originating at the R side, these strings are properly formated M2 code constructed from the inputs to the R functions. That code goes to M2, is evaluated there, and then “exported” with M2’s function toExternalString(). The resulting string often, but not always, produces the M2 code needed to recreate the object resulting from the evaluation, and in that sense is M2’s version of R’s dput(). That string is passed back into R and parsed there into R-style data structures, typically S3-classed lists.

The R-side parsing of the external string from M2 is an expensive process because it is currently implemented in R. Consequently (and for other reasons, too!), in some cases you’ll want to do a M2 computation from R, but leave the output in M2. Since you will ultimately want something in R referencing the result, nearly every m2r function that performs M2 computations has a pointer version. As a simple naming convention, the name of the function that returns the pointer, called the reference function, is determined by the name of the ordinary function, called the value function, by appending a ..

For example, we’ve seen that factor_n() computes the prime decomposition of a number. The corresponding reference function is factor_n.():

(x <- 2^5 * 3^4 * 5^3 * 7^2 * 11^1)
# [1] 174636000
factor_n.(x)
# M2 Pointer Object
#   ExternalString : new Product from {new Power from {2,5},new Power from {3,...
#          M2 Name : m2o460
#         M2 Class : Product (WrapperType)

All value functions simply wrap reference functions and parse the output with m2_parse(), a general M2 parser, often followed by a little more parsing. m2_parse() typically creates an object of class m2 so that R knows what kind of thing it is. For example:

class(factor_n.(x))
# [1] "m2_pointer" "m2"

Even more, m2_parse() often creates objects that have an inheritance structure that references m2 somewhere in the middle of its class structure, with specific structure preceding and general structure succeeding (examples below). Apart from its class, the general principle we follow here for the object itself is this: if the M2 object has a direct analogue in R, it is parsed into that kind of R object and additional M2 properties are kept as metadata (attributes); if there is no direct analogue in R, the object is an NA with metadata.

Perhaps the easiest way to see this is with a matrix. m2_matrix() creates a matrix on the M2 side from input on the R side. In the following, to make things more clear we use magrittr’s pipe operator, with which the following calls are semantically equivalent: g(f(x)) and x %>% f %>% g.

library(magrittr)
mat <- matrix(c(1,2,3,4,5,6), nrow = 3, ncol = 2)
mat %>% m2_matrix.   # = m2_matrix.(mat)
# M2 Pointer Object
#   ExternalString : map((ZZ)^3,(ZZ)^2,{{1, 4}, {2, 5}, {3, 6}})
#          M2 Name : m2rintmatrix00000001
#         M2 Class : Matrix (Type)
mat %>% m2_matrix. %>% m2_parse
#      [,1] [,2]
# [1,]    1    4
# [2,]    2    5
# [3,]    3    6
# M2 Matrix over ZZ[]
mat %>% m2_matrix. %>% m2_parse %>% str
#  'm2_matrix' int [1:3, 1:2] 1 2 3 4 5 6
#  - attr(*, "m2_name")= chr "m2rintmatrix00000003"
#  - attr(*, "m2_meta")=List of 1
#   ..$ ring: 'm2_polynomialring' logi NA
#   .. ..- attr(*, "m2_name")= chr "ZZ"
#   .. ..- attr(*, "m2_meta")=List of 3
#   .. .. ..$ vars    : NULL
#   .. .. ..$ coefring: chr "ZZ"
#   .. .. ..$ order   : chr "grevlex"
mat %>% m2_matrix    # = m2_parse(m2_matrix.(mat))
#      [,1] [,2]
# [1,]    1    4
# [2,]    2    5
# [3,]    3    6
# M2 Matrix over ZZ[]

It may be helpful to think of every m2 object as being a missing value (NA, a logical(1)) with two M2 attributes: their name (m2_name) and a capture-all named list (m2_meta). These can be accessed with m2_name() and m2_meta(). For example, a ring, having no analogous object in R, is an NA with attributes:

r <- ring("x", "y", coefring = "QQ")
str(r)
#  'm2_polynomialring' logi NA
#  - attr(*, "m2_name")= chr "m2rintring00000006"
#  - attr(*, "m2_meta")=List of 3
#   ..$ vars    :List of 2
#   .. ..$ : chr "x"
#   .. ..$ : chr "y"
#   ..$ coefring: chr "QQ"
#   ..$ order   : chr "grevlex"
class(r)
# [1] "m2_polynomialring" "m2"
m2_name(r)
# [1] "m2rintring00000006"
m2_meta(r)
# $vars
# $vars[[1]]
# [1] "x"
# 
# $vars[[2]]
# [1] "y"
# 
# 
# $coefring
# [1] "QQ"
# 
# $order
# [1] "grevlex"

But a matrix of integers isn’t:

mat <- m2_matrix(matrix(c(1,2,3,4,5,6), nrow = 3, ncol = 2))
str(mat)
#  'm2_matrix' num [1:3, 1:2] 1 2 3 4 5 6
#  - attr(*, "m2_name")= chr "m2rintmatrix00000005"
#  - attr(*, "m2_meta")=List of 1
#   ..$ ring: 'm2_polynomialring' logi NA
#   .. ..- attr(*, "m2_name")= chr "ZZ"
#   .. ..- attr(*, "m2_meta")=List of 3
#   .. .. ..$ vars    : NULL
#   .. .. ..$ coefring: chr "ZZ"
#   .. .. ..$ order   : chr "grevlex"
class(mat)
# [1] "m2_matrix" "m2"        "matrix"
m2_name(mat)
# [1] "m2rintmatrix00000005"
m2_meta(mat)
# $ring
# M2 Ring: ZZ[], grevlex order

Since a matrix of integers is an object in R, it’s represented as one, and consequently we can compute with it directly as it if it were a matrix; it is. On the other hand, since a ring is not, it’s an NA. When dealing with M2, object like rings, that is to say objects without R analogues, are more common than those like integer matrices.

Creating your own m2r wrapper

We’ve already wrapped a number of Macaulay2 functions; for a list of functions in m2r, check out ls("package:m2r"). But the list is very far from exhaustive. To create your own wrapper function of a Macaulay2 command, you’ll need to create an R file that looks like the one below. This will create both value (e.g. f) and reference/pointer (e.g. f.) versions of the function. As a good example of these at work, see the scripts for factor_n() or factor_poly().

#' Function documentation header
#'
#' Function header explanation, can run several lines. Function
#' header explanation, can run several lines. Function header
#' explanation, can run several lines.
#'
#' @param esntl_parm_1 esntl_parm_1 description
#' @param esntl_parm_2 esntl_parm_2 description
#' @param code return only the M2 code? (default: \code{FALSE})
#' @param parse_parm_1 parse_parm_1 description
#' @param parse_parm_2 parse_parm_2 description
#' @param ... ...
#' @name f
#' @return (value version) parsed output or (reference/dot version)
#'   \code{m2_pointer}
#' @examples
#'
#' \dontrun{ requires Macaulay2 be installed
#'
#' # put examples here
#' 1 + 1
#'
#' }
#'

# value version of f (standard user version)
#' @rdname f
#' @export
f <- function(esntl_parm_1, esntl_parm_2, code = FALSE, parse_parm_1, parse_parm_2, ...) {

  # run m2
  args <- as.list(match.call())[-1]
  eargs <- lapply(args, eval, envir = parent.frame())
  pointer <- do.call(f., eargs)
  if(code) return(invisible(pointer))

  # parse output
  parsed_out <- m2_parse(pointer)

  # more parsing, like changing classes and such
  TRUE

  # return
  TRUE

}

# reference version of f (returns pointer to m2 object)
#' @rdname f
#' @export
f. <- function(esntl_parm_1, esntl_parm_2, code = FALSE, ...) {

  # basic arg checking
  TRUE

  # create essential parameters to pass to m2 this step regularizes input to m2, so it
  # is the one that deals with pointers, chars, rings, ideals, mpolyLists, etc.
  TRUE

  # construct m2_code from regularized essential parameters
  TRUE

  # message
  if(code) { message(m2_code); return(invisible(m2_code)) }

  # run m2 and return pointer
  m2.(m2_code)

}

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant Nos. 1321794 and 1622449.

Installation

Here’s how you can install the current developmental version of m2r.

if (!requireNamespace("devtools")) install.packages("devtools")
devtools::install_github("coneill-math/m2r")

For m2r to find Macaulay2, you’ll need to set an environmental variable in your ~/.Renviron file. To do that, run this:

if (!requireNamespace("usethis")) install.packages("usethis")
usethis::edit_r_environ()

And, in the text file that opens, add a line such as M2=/Applications/Macaulay2-1.10/bin.